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Theorem fores 5036
Description: Restriction of a function. (Contributed by NM, 4-Mar-1997.)
Assertion
Ref Expression
fores ((Fun 𝐹 A ⊆ dom 𝐹) → (𝐹A):Aonto→(𝐹A))

Proof of Theorem fores
StepHypRef Expression
1 funres 4863 . . 3 (Fun 𝐹 → Fun (𝐹A))
21anim1i 323 . 2 ((Fun 𝐹 A ⊆ dom 𝐹) → (Fun (𝐹A) A ⊆ dom 𝐹))
3 df-fn 4828 . . 3 ((𝐹A) Fn A ↔ (Fun (𝐹A) dom (𝐹A) = A))
4 df-ima 4281 . . . . 5 (𝐹A) = ran (𝐹A)
54eqcomi 2022 . . . 4 ran (𝐹A) = (𝐹A)
6 df-fo 4831 . . . 4 ((𝐹A):Aonto→(𝐹A) ↔ ((𝐹A) Fn A ran (𝐹A) = (𝐹A)))
75, 6mpbiran2 834 . . 3 ((𝐹A):Aonto→(𝐹A) ↔ (𝐹A) Fn A)
8 ssdmres 4556 . . . 4 (A ⊆ dom 𝐹 ↔ dom (𝐹A) = A)
98anbi2i 433 . . 3 ((Fun (𝐹A) A ⊆ dom 𝐹) ↔ (Fun (𝐹A) dom (𝐹A) = A))
103, 7, 93bitr4i 201 . 2 ((𝐹A):Aonto→(𝐹A) ↔ (Fun (𝐹A) A ⊆ dom 𝐹))
112, 10sylibr 137 1 ((Fun 𝐹 A ⊆ dom 𝐹) → (𝐹A):Aonto→(𝐹A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1226  wss 2890  dom cdm 4268  ran crn 4269  cres 4270  cima 4271  Fun wfun 4819   Fn wfn 4820  ontowfo 4823
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-res 4280  df-ima 4281  df-fun 4827  df-fn 4828  df-fo 4831
This theorem is referenced by:  resdif  5069
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